Can a single black hole emit gravitational waves and evaporate?

Question

I have read this:

Yes, single neutron stars can emit gravitational waves if they have sufficient asymmetries. For some background, an object symmetric about its axis of rotation does not produce gravitational waves. A quote from Hartle's An Introduction to Einstein's General Relativity in the example "A Little Rotation" on page 497 of the textbook: Axisymmetric rotation in general is an example of a highly symmetric motion that does not produce gravitational radiation. For clarity, "axisymmetric" means "symmetric about an axis." To view the details of the calculation, that section of Hartle is a good resource.However, a rotating neutron star which does not have sufficient symmetry would emit gravitational waves. What's more, as pointed out by @Harti, pulsars must exhibit some sort of asymmetry in order to emit the radiation they do - but the question is whether these asymmetries are such that they will produce detectable gravitational waves.

Can gravitational waves be emitted from single neutron stars?

Now let's assume the case where this specific neutron star (that is already emitting GWs) collapses into a black hole. Very naively thinking, the asymmetric rotation remains after the collapse, so the black hole will emit GWs too. Now would this be considered a way for the black hole to evaporate? GWs carry energy away from the black hole ('s gravitational field's energy), so they should reduce it's mass. Since this reduces its mass (just like Hawking radiation), this gravitational radiation could lead to evaporation (if very simply the GW energy is more than the CMB, and there is no inflow of matter/energy into the black hole).

Question:

1. Can a single black hole emit gravitational waves and evaporate?

Answer 1

Árpád Szendrei wrote: "Very naively thinking, the asymmetric rotation remains after the collapse, so the black hole will emit GWs too."

After the ringdown axial symmetry is achieved and no more gravitational waves are emitted. Thus the no hair theorem which also excludes quadrupole, so the horizon should asymptotically converge into a perfect rotational ellipsoid.

Árpád Szendrei asked: "The black hole will emit GWs too. Now would this be considered a way for the black hole to evaporate?"

Gravitational waves can't reduce the rest mass of a particle or a black hole, they only reduce the kinetic energy of the components of a system which also contributes to its gravity (if you heat up a box of water with thermal energy E it will get heavier by ΔM=E/c²). At the very least the irreducible mass must remain (just like the water won't lose all its weight by cooling down).